THE PENNSYLVANIA STATE UNIVERSITY SCHREYER HONORS COLLEGE DEPARTMENT OF ENGINEERING SCIENCE AND MECHANICS AND SCHOOL OF MUSIC MODELING INTONATION IN NON-WESTERN MUSICAL CULTURES SAMUEL LAPP SPRING 2017 A thesis submitted in partial fulfillment of the requirements for a baccalaureate degree in Engineering Science with interdisciplinary honors in Engineering Science and Music Reviewed and approved* by the following: Matthew B. Parkinson Associate Professor of Engineering Design and Mechanical Engineering Thesis Supervisor Mark Ballora Associate Professor of Music Technology Honors Adviser Gary Gray Associate Professor of Engineering Science and Mechanics Honors Adviser Judith A. Todd Department Head P. B. Breneman Chair and Professor of Engineering Science and Mechanics *Signatures are on file in the Schreyer Honors College and Engineering Science and Mechanics Office. i Abstract Musical cultures around the world use tuning systems very different from the tuning system used in Western music. Comparing and contrasting intonation from various cultures could give insight into the question of learned versus innate preferences in the perception of music, and into the cultural development of musical traditions. However, methods for analyzing intonation in Western music have not been generalized to non-Western contexts. Flexible and non-discrete tuning used in cultures such as Arabic maqam music pose a chal- lenge for analyzing intonation. This paper attempts to generalize Stolzenburg’s “Periodicity” method and Gill and Purves’ “Similarity” method for microtonal music, and finds that they are ill-suited for the analysis of Arabic music. An alternative approach is developed, which analyzes the relative intonation of a series of consecutive notes. This approach reveals that Arabic intonation is fundamentally different from Western and Chinese intonation. While Chinese and Western intonation are based on intervals from a fixed scale, Arabic intonation is based on a continuous spectrum of small intervals. ii Table of Contents List of Figures v List of Tables vii Acknowledgements viii I Background 3 1 Intonation in Western, Arabic, and Chinese Cultures 4 1.1 Three Perspectives of Intonation ........................ 4 1.2 Western Music .................................. 5 1.2.1 scales ................................... 5 1.2.2 chords and harmony ........................... 6 1.3 Arabic Music ................................... 7 1.3.1 instruments used in Arabic music .................... 9 1.4 Chinese Music ................................... 9 2 Harmony: Studies of Consonance and Dissonance 11 2.1 Computational Methods for Harmony Analysis ................ 12 2.2 History of Consonance and Dissonance Analysis in Western Music ...... 12 2.3 Perception of Intervals of Simple and Complex Tones ............. 13 2.3.1 simple and complex tones ........................ 13 2.3.2 dissonance of dyads ............................ 14 2.4 Factors Influencing Consonance and Dissonance ................ 15 2.4.1 pleasantness versus consonance ..................... 17 2.4.2 multiple dimensions of harmony ..................... 17 3 Two Models for Predicting Common Scales 18 3.1 Periodicity Model for Dissonance ........................ 18 3.1.1 method, claims and results ........................ 20 3.1.2 generalizing to microtonal ........................ 23 3.1.3 analysis .................................. 23 3.2 Similarity Model for Dissonance ......................... 23 3.2.1 methods, claims and results ....................... 23 3.2.2 recreating the similarity algorithm ................... 26 3.2.3 analysis .................................. 27 iii II Methods 28 4 Developing a Model to Analyze Intonation 29 4.1 An Algorithm for Analyzing Music ....................... 29 4.1.1 finding the fundamental frequency ................... 29 4.1.2 subdivision of notes ........................... 30 4.1.3 sample length and frequency resolution ................. 32 4.1.4 parameters and optimization ...................... 32 4.2 Testing the Model ................................ 32 4.2.1 monophonic and polyphonic music ................... 33 4.2.2 from pitches back to music ........................ 34 5 Applying the Tools to Periodicity and Similarity 37 5.1 Extracting Scales ................................. 37 5.2 Characterizing Maqamat ............................. 38 5.3 Periodicity and Similarity of Maqamat ..................... 40 6 Non-Discretized Intonation 43 6.1 Pitch Distribution and Scatter .......................... 43 6.1.1 case study: the neutral third ...................... 43 6.1.2 case study: the leading tone in Western music ............. 44 6.1.3 issues with fixed pitch set ........................ 46 6.2 Shape Grammars: Words and Vocabularies ................... 46 6.2.1 finding words ............................... 46 6.2.2 vocabulary ................................ 47 III Results, Analysis, and Conclusions 48 7 Accuracy of Vocabulary Extraction 49 7.1 Tempo and Polyphony .............................. 49 7.1.1 slow, monophonic music ......................... 49 7.1.2 fast, polyphonic music .......................... 51 7.1.3 file formats ................................ 52 7.2 Symmetry of Three-Syllable Words ....................... 54 8 Comparing Vocabularies 56 8.1 Vocabulary of Composers ............................ 56 8.2 Vocabulary of a Genre .............................. 56 9 Vocabularies Across Cultures 60 9.1 Intervals and Words ............................... 60 9.2 Vocabulary of Three Genres ........................... 60 9.3 Returning to Periodicity ............................. 62 iv 10 Conclusions 66 10.1 Words as Building Blocks ............................ 66 10.2 Discrete Versus Continuous Measurement Systems ............... 67 10.2.1 aural versus written traditions ...................... 67 10.2.2 uncertainty: a quantum physics analogy ................ 68 10.3 The Origins of Scales ............................... 68 10.3.1 Music Is More Than Pitch ........................ 68 10.4 Further Research ................................. 69 Appendix A Human Hearing: From Pressure Waves to Neural Signals 72 Appendix B Whole Number Ratio Algorithm 77 Appendix C Works Cited: Music Recordings 78 Appendix D Listening Tracks 79 Bibliography 80 v List of Figures 1.1 C Major Scale ................................... 6 1.2 Western Modes .................................. 7 1.3 Three Maqamat in Western Notation ...................... 8 1.4 Maqam Bayati .................................. 8 1.5 Chinese Pentatonic Scales ............................ 9 1.6 Chinese Twelve Tone Tuning ........................... 10 2.1 Illustration of Beating .............................. 13 2.2 Frequency Response of a Violin ......................... 14 2.3 Dissonance of Simple Dyads ........................... 15 2.4 Dissonance of Complex Dyads .......................... 16 3.1 Illustration of the Periodicity Method ...................... 19 3.2 Illustration of the Similarity Method ...................... 25 4.1 Root-Finding Algorithm: False-Positive ..................... 30 4.2 Transient Frequency Response .......................... 31 4.3 Monophony and Polyphony ........................... 33 4.4 Transcribing a Slow, Monophonic Piece ..................... 34 4.5 Transcribing a Fast, Polyphonic Piece ...................... 35 5.1 Histogram of Frequencies in Beethoven ..................... 38 5.2 Notation of Maqam Nahawand ......................... 39 5.3 Pitch Histogram of an Arabic Piece ....................... 39 5.4 Comparision of Tunings for Maqam Rast .................... 40 7.1 Common Words in Bach’s Music ........................ 50 7.2 Vocabulary Extraction in a Short Monophonic Selection ........... 51 7.3 Vocabulary Extraction in a Short Polyphonic Selection ............ 52 7.4 Vocabulary in Two Recordings of Beethoven .................. 53 7.5 Vocabulary Extraction and File Format ..................... 54 8.1 Vocabulary in Bach’s Music ........................... 57 8.2 Vocabulary in Bach and Beethoven ....................... 57 8.3 Vocabulary in Brahms and Beethoven ...................... 58 8.4 Intervals in Western Music ............................ 59 9.1 Common Western and Arabic Words ...................... 61 9.2 Comparing Vocabularies of Three Genres .................... 63 9.3 Similarity and Periodicity of Common Words ................. 65 vi A.1 Hearing Streams of Audio ............................ 73 A.2 Parts of the Ear .................................. 73 A.3 Sound as Pressure Waves ............................. 74 A.4 Basilar Membrane ................................ 75 vii List of Tables 1.1 Western Intervals ................................. 5 2.1 Consonant and Dissonant Intervals ....................... 12 3.1 Periodicity of Triads ............................... 21 3.2 Ranking Heptatonic Scales with Periodicity .................. 22 3.3 Ranking Heptatonic Scales with Microtonal Periodicity ............ 24 3.4 Ranking Heptatonic Scales with Similarity ................... 26 5.1 Maqam Periodicity Rankings .......................... 41 5.2 Periodicity and Similarity of Maqamat ..................... 42 6.1 Failure of Periodicity Method with Small Tolerance .............. 44 6.2 Most Common Frequencies in Bach’s Music .................. 45 9.1 Ranking Words with Periodicity and Similarity ................ 64 viii Acknowledgements There are many people to thank for their generous support throughout this research. First, my thesis